What is the ruling regarding my mother praying sitting down on the train when she is able to pray standing, but she does not wish to appear before men, knowing that she delays Fajr prayer until its time has passed so she can pray it standing? And what is the criterion for inability to stand, given the possibility of finding a place in other carriages or the available place being dirty?

It is forbidden for a mother to delay prayer until its time has passed. She must pray standing if she is able, otherwise, she may pray sitting. One must search for a suitable place to pray standing on a train, and praying sitting is not valid unless one is unable to stand due to obvious hardship. If a clean place is found for praying standing, it is obligatory to do so. A prayer mat can be used to avoid dust and dirt. The questioner and his mother must make up the prayers that were performed without standing, despite being able to do so. If the number of these prayers is unknown, they must continue to make them up until they are reasonably certain that their obligation has been fulfilled. The preponderant opinion is that of the majority of scholars, which states that making up these prayers is obligatory.